Getting Triggy With It
|
|
- Wilfrid Morris
- 5 years ago
- Views:
Transcription
1 Getting Triggy With It Date: 15 May 2013 Topic: Pythagorean Theorem and Trigonometric Ratios Class: Grade 9 Ability Level: Mixed Ability Teacher: Mr. Cyrus Alvarez LESSON OBJECTIVES: At the end of the lesson, the students should be able to: 1. Explain the characteristics of a right triangle and correctly identify its parts; 2. Explain the relationships between the different parts of the triangle; 3. Understand and correctly apply the Pythagorean Theorem and the three primary trigonometric ratios in solving right triangles. 4. Investigate applications of the Pythagorean Theorem and the three primary trigonometric ratios in real-life situations. PRE-REQUISITES: 1. Students have mastery over the concepts of angles, measurements, ratio and proportion, exponents, radicals, and solutions to algebraic equations. 2. Students understand the basic properties of triangles, such as the sum of its interior angles, triangle inequality, similarity and congruence, etc. 3. Students should be familiar with the use of scientific calculators or similar technologies. FOCUS: Investigative activity Cooperative Learning Problem Solving ACTIVITY The teacher uses the following situation as an activation strategy: You are an engineer commissioned to design a fantype cable-stayed foot bridge with one tower in the middle and 5 symmetric pairs of wire cables attached on top of the tower. The bridge is supposed to cover a length of 240 meters and the guy wires are supposed to be attached on the bridge at equal intervals. If the longest pair of wire cable has to be attached at a 30 angle from the ends of the bridge deck, how tall should the tower be? What is the total length of cable wires needed for the bridge? RESOURCES/MATERIALS Source: 5/most-famous-bridges-world /
2 The teacher explains that the tower and the cable wire create a right triangle with the bridge deck, thus the properties of right triangles may be used to solve the problems presented in the situation. The teacher asks about ideas regarding right triangles. Establish understanding that right triangles are triangles with one angle measuring 90 o, and the two other angles are acute and complementary. The teacher builds the students vocabulary with the following parts of a right triangle: Hypotenuse the longest side of a right triangle, which is opposite the right angle of a right triangle. Legs the two shorter sides of a right triangle. Complementary Angles angles whose measures add up to 90 o. The two acute angles of a right triangle are complementary. Adjacent Side the leg of a right triangle that forms the angle with the hypotenuse. Opposite Side the leg of a right triangle that is not part of the angle, but is opposite the angle. The teacher divides students into groups of 3-4 students and provides the Right Triangles Worksheetfor each group to Establish understanding of the Pythagorean Theorem as the relationship of the length of the hypotenuse with the lengths of the legs (a 2 + b 2 = c 2, where a and b are the legs and c is the hypotenuse). Establish understanding that in similar right triangles (i.e, right triangles whose angles have the same measures even if the sides do not have the same lengths), the ratio between two sides will always be the same. The teacher gives minutes for the groups to work on the worksheet. The teacher moves around to check if: 1) the students are on track 2) all members are doing the said task A cable-stayed bridge has one or more towers where support wire cables are attached from the bridge deck. A fan-type cable stayed bridge is one where the wire cables are connected to or passed over the top tower towards several points on the bridge deck. Worksheet 1 (attached) Materials: paper, scissors, protractor and ruler Technological requirement:scientific calculator, Geogebra*, laptop*, digital camera*, and LCD projector* * if available
3 3) if technology is available, the teacher may take pictures of the students responses for easy presentation The teacher calls on the group to share their findings/answers to the worksheet questions. Possible answers to worksheet questions: 1.1. The hypotenuse decreases as well The ratios remain the same The measures of the acute angles of a right triangle depend on the ratios of its sides They are equal The sum of the squares of the legs is equal to the square of the hypotenuse. (Alternatively, if resources are available, the teacher may use Worksheet 1A, which utilizes Geogebra for the activity.) The teacher defines the three primary trigonometric ratios. sin θ = cos θ = tan θ = Opposite Side Hypotenuse Adjacent Side Hypotenuse Opposite Side Adjacent Side Provide mnemonic for easier recall: SOH-CAH-TOA The teacher defines the Pythagorean Theorem. a 2 + b 2 = c 2, where a and b are legs, and c is the hypotenuse The teacher provides sample problems: Given: Source: Barnett, R. et al, 2008 Technological requirement: Scientific calculator Find: (a) The hypotenuse (25) (b) sin θ ( 7 25 ) (c) cos θ ( ) (d) tan θ ( 7 24 ) Using the figure and the quantities given, find the other quantities: (a) β = 17.8 o, c = 3.45 (α = 72.2 o, a = 3.28, b = 1.05) (b) α = 23 o, a = 54.0 (β = 67 o, b = 127.2, c = 138.2) (c) β = 43 o 20, a =123 (α = 46 o 40, b = 116, c = 169) (d) α = o, b = (β = 36 o 39, a = 32.02, c = 39.90)
4 The teacher goes back to the student groups and instructs them to come up with solutions on the original problem posted at the beginning of the lesson. The Fan-Type Cable-Stayed Bridge Worksheet is distributed among the groups as guide. Solutions, complete with illustrations and calculations, are to be written on a piece of cartolina for presentation. The teacher moves around to check if: 1) the students are on track 2) all members are doing the said task 3) if technology is available, the teacher may take pictures of the students responses for easy presentation The teacher gives minutes for the groups to work on the worksheet. Worksheet 2 (attached) Materials: Ruler, protractor, white cartolina and markers Technological Requirement: Scientific calculator, laptop*, digital camera*, and LCD projector* * if available The teacher calls on the group to share their solutions. Sample illustration and calculation: The teacher summarizes lesson by playing the Gettin Triggy With It music video to emphasize the following points: 1. A right triangle is a triangle with one angle measuring 90 o and the other two angles are acute and complementary. 2. The sides of a right triangle are related to each other via the Pythagorean Theorem. 3. The acute angles of a right triangle are dependent on the ratios of its sides. 4. These ratios are called trigonometric ratios. The three primary trigonometric ratios are sine, cosine and tangent.the sine of an angle is the ratio of the opposite side and the hypotenuse; the cosine of an angle is the Getting Triggy With It music video v=t2upyylh4zo
5 ratio of the adjacent side and the hypotenuse; while the tangent of an angle is the ratio of the opposite side and the adjacent side. 5. In a right triangle, if at least two quantities are known, the other quantities may be derived using Pythagorean Theorem and/or the trigonometric ratios. The teacher gives the following homework to the students: 1. Journal/blog entry on their journals/cms pages about what they have learned from the session and their insights on how this new knowledge can be applied in their lives. Students should also include answers to the following questions: a. What specific topics and/or skills helped you accomplish the activities? b. What made the tasks challenging? What are the difficulties that you encountered while doing the tasks? c. How can you improve on the difficulties you encountered? 2. Pair up students to work together on the Applications of Right Triangles Worksheet. 3. Divide students into groups of 5 and assign each group to come up with and film a unique music video that deals with the concepts discussed in this lesson. Worksheet 3 (attached) Evaluation/Assessment:Worksheet and class participation, both will be assessed using the Classwork/Participation Rubric and the Peer Evaluation Rubric (for group work) generated from irubric References: Aufmann, Richard N., Barker, Vernon C. and Nation, Richard D. (2011). College Algebra and Trigonometry, 11 th ed. Barnett, Raymond A. et al (2008). College Algebra with Trigonometry, 9 th ed. Larson, Ron (2012). Algebra and Trigonometry: Real Mathematics, Real People, 6 th ed. McKeague, Charles P. and Turner, Mark D. (2008). Trigonometry, 7th ed. Sullivan, Michael and Sullivan, Michael III (2009). Algebra & Trigonometry, 6 th ed.
6 Worksheet 1: Right Triangles Group No.: Date: Members: Materials: paper, scissors, protractor, ruler, and scientific calculator β Procedure: 1. Cut a right triangle of any size from the piece of paper. 2. Identify the hypotenuse of the right triangle as side c, and the legs as sides a and b, respectively. a c 3. Measure the sides of the right triangle using the ruler and record your measurements on Table 1. Use centimeters as your unit of measurement. 4. Label the angle opposite leg a as α and the angle opposite leg b as β. 5. Measure the angles using the protractor and record your measurements on Table Cut the right triangle through a line parallel to one b side. Make sure that angles α and β remains the same. β 7. Measure the sides of the new triangle and record the measurements on Table 1. a c 8. Repeat (6) and (7). 9. Compute for the ratios as indicated in Table Answer the questions that follow. 11. Using the values in Table 1, compute for the values as indicated in Table 2. α 12. Answer the questions that follow. b Table 1 α Triangle 1 Triangle 2 Triangle 3 a b c α β a/c b/c a/b b Questions based on Table 1: 1. When the lengths of the legs of the triangle are decreased, what happens to the length of the hypotenuse? 2. What do you notice about the ratio of the length of leg a to the hypotenuse c in the three triangles measured, given that the angles remain the same? How about b and c? How about a and b?
7 3. What conclusions can you draw regarding the ratios of the sides of a right triangle relative to its angles? Table 2 Triangle 1 Triangle 2 Triangle 3 a 2 b 2 a 2 + b 2 c 2 Questions based on Table 2: 1. What do you notice about the sum of the squares of the legs of the right triangles in relation to the square of its hypotenuse? 2. What conclusion can you draw regarding the relationship between the lengths of the legs of a right triangle with the length of its hypotenuse?
8 Worksheet 1A: Right Triangles Group No.: Date: Members: Technological Requirements: Software: Geogebra ( ) Hardware:Computer, Scientific Calculator Procedure: 1. Click on Angle with Given Size. Click on the origin, then click on any point on the x-axis. Provide any angle between 0 o and 90 o, and choose clockwise.
9 2. Define a ray going from point B to point A. Label the ray as l. Hide the point A.
10 3. Click on Segment between Two Points. Click on Point A, then on Point B. Click again on Point A and then click on the intersection between the y-axis and the ray l. Finally, click on Point B and then click on the intersection between the y-axis and the ray l. Show the labels on these 3 line segments. Hide the ray l.
11 4. Record the values of a, b and c on the table below, as well as the measure of α. Then, compute for β = 90 o α. 5. Drag the Point B on any positive point on the x-axis. Record the values of a, b and c on the table. Do this again three times so that you will have 5 sets of data. 6. Compute for the ratios indicated on Table 1 (i.e., a c, b c, and a b ). 7. Compute for the quantities indicated on Table 2 as well. 8. Answer the questions that follow. Table 1 a b c α β a/c b/c a/b Triangle 1 Triangle 2 Triangle 3 Triangle 4 Triangle 5 Questions based on Table 1: 1. When the lengths of the legs of the triangle are decreased, what happens to the length of the hypotenuse? 2. What do you notice about the ratio of the length of leg a to the hypotenuse c in the three triangles measured, given that the angles remain the same? How about b and c? How about a and b? _ 3. What conclusions can you draw regarding the measures of the acute angles of a right triangle in relation to its sides? Triangle 1 Triangle 2 Triangle 3 Triangle 4 Triangle 5 Table 2 a 2 b 2 a 2 + b 2 c 2 Questions based on Table 2: 1. What do you notice about the sum of the squares of the legs of the right triangles in relation to the square of its hypotenuse? 2. What conclusion can you draw regarding the relationship between the lengths of the legs of a right triangle with the length of its hypotenuse?
12 Worksheet 2: Fan-Type Cable-Stayed Bridge Group No.: Date: Members: Materials:Ruler, protractor, white cartolina, markers, and scientific calculator Situation: You are a group of engineers hired to designa fan-type cable-stayed foot bridge. A cable-stayed bridge has one or more towers where support wire cables are attached from the bridge deck. A fan-type cable-stayed bridge is one where the wire cables are connected to or passed over the top tower towards several points on the bridge deck. You are expected to come up with a proposal for your bridge design. Afterwards, you will present your findings to the CEO (your teacher) of the Pythagoras Towers Company. Problem: The fan-type cable-stayed foot bridge that you re designing will have one tower in the middle and 5 symmetric pairs of wire cables attached on top of the tower. The bridge is supposed to cover a length of 240 meters and the guy wires are supposed to be attached on the bridge at equal intervals. If the longest pair of wire cable has to be attached at a 30 angle from the ends of bridge deck, how tall should the tower be? What is the total length of cable wires needed for the bridge? Requirements: 1. Create a proposal for the required lengths of guy wire needed to build the two towers given the specifications. Make sure that you have clear illustrations and calculations to substantiate your proposal. 2. Write your illustrations and complete solutions on the cartolina for presentation. 3. Answer the questions that follow. Questions 1. Suppose that the contractor wanted to save some money and decided to reduce the number of support cables from 5 pairs to 4 pairs, which will still be all equally spaced throughout the bridge, how much cable wire will be saved? 2. If you can put the tower ANYWHERE on the bridge, where will you put it so that you can have the least amount of cable wires to use, considering that there should be a total of 10 cable wires attached uniformly on the bridge deck and the height of the tower remains the same?
13 Worksheet 3: Applications of Right Triangles Names: Date: Technological Requirements: Hardware: Scientific Calculator Procedure: 1. Write down two real-life situation problems that would involve the use of right triangles to solve. Make sure that your problem is original and not copied from anywhere. PROBLEM No. 1 PROBLEM No. 2
14 Names: 2. Exchange questions with another pair (Page 1 of this worksheet). DO NOT show them your solutions. Write the names of the pair you exchanged questions with here:. And write down your solutions to their questions here.
15 Names: 3. Write down the solution for your problems here.
16 Names: 4. Compare your solutions with the other pair s solutions. 5. Answer the following questions: a. Did both your groups get the same answers for your problems? a.1. If not, who got it right? How did someone get the wrong answer? b. Did you use the same steps to solve your problem? b.1. If not, which one is the easier or faster solution? Why? c. Are there any other possible solutions that may be easier or faster? If so, what is it? d. Are the other pair s problems too easy or too difficult? Why do you say so? e. What are your suggestions to improve the statement of their problem?
Lesson 27: Sine and Cosine of Complementary and Special Angles
Lesson 7 M Classwork Example 1 If α and β are the measurements of complementary angles, then we are going to show that sin α = cos β. In right triangle ABC, the measurement of acute angle A is denoted
More informationChapter 2: Pythagoras Theorem and Trigonometry (Revision)
Chapter 2: Pythagoras Theorem and Trigonometry (Revision) Paper 1 & 2B 2A 3.1.3 Triangles Understand a proof of Pythagoras Theorem. Understand the converse of Pythagoras Theorem. Use Pythagoras 3.1.3 Triangles
More informationThe Pythagorean Theorem
. The Pythagorean Theorem Goals Draw squares on the legs of the triangle. Deduce the Pythagorean Theorem through exploration Use the Pythagorean Theorem to find unknown side lengths of right triangles
More informationTrigonometry. An Overview of Important Topics
Trigonometry An Overview of Important Topics 1 Contents Trigonometry An Overview of Important Topics... 4 UNDERSTAND HOW ANGLES ARE MEASURED... 6 Degrees... 7 Radians... 7 Unit Circle... 9 Practice Problems...
More informationLesson Idea by: Van McPhail, Okanagan Mission Secondary
Click to Print This Page Fit by Design or Design to Fit Mechanical Drafter Designer Lesson Idea by: Van McPhail, Okanagan Mission Secondary There's hardly any object in your home or school that hasn't
More informationHow to work out trig functions of angles without a scientific calculator
Before starting, you will need to understand how to use SOH CAH TOA. How to work out trig functions of angles without a scientific calculator Task 1 sine and cosine Work out sin 23 and cos 23 by constructing
More informationHow to Do Trigonometry Without Memorizing (Almost) Anything
How to Do Trigonometry Without Memorizing (Almost) Anything Moti en-ari Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ c 07 by Moti en-ari. This work is licensed under the reative
More informationThe Pythagorean Theorem and Right Triangles
The Pythagorean Theorem and Right Triangles Student Probe Triangle ABC is a right triangle, with right angle C. If the length of and the length of, find the length of. Answer: the length of, since and
More informationMathematics Geometry Grade 6AB
Mathematics Geometry Grade 6AB It s the Right Thing Subject: Mathematics: Geometry: Ratio and Proportion Level: Grade 7 Abstract: Students will learn the six types of triangles and the characteristics
More informationOne of the classes that I have taught over the past few years is a technology course for
Trigonometric Functions through Right Triangle Similarities Todd O. Moyer, Towson University Abstract: This article presents an introduction to the trigonometric functions tangent, cosecant, secant, and
More informationMay 03, AdvAlg10 3PropertiesOfTrigonometricRatios.notebook. a. sin17 o b. cos 73 o c. sin 65 o d. cos 25 o. sin(a) = cos (90 A) Mar 9 10:08 PM
a. sin17 o b. cos 73 o c. sin 65 o d. cos 25 o sin(a) = cos (90 A) Mar 9 10:08 PM 1 Find another pair of angle measures x and y that illustrates the pattern cos x = sin y. Mar 9 10:11 PM 2 If two angles
More informationMarch 29, AdvAlg10 3PropertiesOfTrigonometricRatios.notebook. a. sin17 o b. cos 73 o c. sin 65 o d. cos 25 o. sin(a) = cos (90 A) Mar 9 10:08 PM
a. sin17 o b. cos 73 o c. sin 65 o d. cos 25 o sin(a) = cos (90 A) Mar 9 10:08 PM 1 Find another pair of angle measures x and y that illustrates the pattern cos x = sin y. Mar 9 10:11 PM 2 If two angles
More informationSpecial Right Triangles and Right Triangle Trigonometry
Special Right Triangles and Right Triangle Trigonometry Reporting Category Topic Triangles Investigating special right triangles and right triangle trigonometry Primary SOL G.8 The student will solve real-world
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar
More informationCreate Your Own Triangles Learning Task
Create Your Own Triangles Learning Task Supplies needed Heavy stock, smooth unlined paper for constructing triangles (unlined index cards, white or pastel colors are a good choice) Unlined paper (if students
More informationChapter 1 and Section 2.1
Chapter 1 and Section 2.1 Diana Pell Section 1.1: Angles, Degrees, and Special Triangles Angles Degree Measure Angles that measure 90 are called right angles. Angles that measure between 0 and 90 are called
More informationChapter 11 Trigonometric Ratios The Sine Ratio
Chapter 11 Trigonometric Ratios 11.2 The Sine Ratio Introduction The figure below shows a right-angled triangle ABC, where B = and C = 90. A hypotenuse B θ adjacent side of opposite side of C AB is called
More informationUnit 5. Algebra 2. Name:
Unit 5 Algebra 2 Name: 12.1 Day 1: Trigonometric Functions in Right Triangles Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Theta Opposite Side of an Angle Adjacent Side of
More informationT.2 Trigonometric Ratios of an Acute Angle and of Any Angle
408 T.2 Trigonometric Ratios of an Acute Angle and of Any Angle angle of reference Generally, trigonometry studies ratios between sides in right angle triangles. When working with right triangles, it is
More informationName: A Trigonometric Review June 2012
Name: A Trigonometric Review June 202 This homework will prepare you for in-class work tomorrow on describing oscillations. If you need help, there are several resources: tutoring on the third floor of
More information13.4 Chapter 13: Trigonometric Ratios and Functions. Section 13.4
13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 1 13.4 Chapter 13: Trigonometric Ratios and Functions Section 13.4 2 Key Concept Section 13.4 3 Key Concept Section 13.4 4 Key Concept Section
More informationIntroduction to Trigonometry. Algebra 2
Introduction to Trigonometry Algebra 2 Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta
More informationChapter 3, Part 1: Intro to the Trigonometric Functions
Haberman MTH 11 Section I: The Trigonometric Functions Chapter 3, Part 1: Intro to the Trigonometric Functions In Example 4 in Section I: Chapter, we observed that a circle rotating about its center (i.e.,
More informationPrerequisite Knowledge: Definitions of the trigonometric ratios for acute angles
easures, hape & pace EXEMPLAR 28 Trigonometric Identities Objective: To explore some relations of trigonometric ratios Key Stage: 3 Learning Unit: Trigonometric Ratios and Using Trigonometry Materials
More informationRight Triangle Trigonometry (Section 4-3)
Right Triangle Trigonometry (Section 4-3) Essential Question: How does the Pythagorean Theorem apply to right triangle trigonometry? Students will write a summary describing the relationship between the
More informationPreCalculus 4/10/13 Obj: Midterm Review
PreCalculus 4/10/13 Obj: Midterm Review Agenda 1. Bell Ringer: None 2. #35, 72 Parking lot 37, 39, 41 3. Homework Requests: Few minutes on Worksheet 4. Exit Ticket: In Class Exam Review Homework: Study
More information5/6 Lesson: Angles, measurement, right triangle trig, and Pythagorean theorem
5/6 Lesson: Angles, measurement, right triangle trig, and Pythagorean theorem I. Lesson Objectives: -Students will be able to recall definitions of angles, how to measure angles, and measurement systems
More informationDouble-Angle, Half-Angle, and Reduction Formulas
Double-Angle, Half-Angle, and Reduction Formulas By: OpenStaxCollege Bicycle ramps for advanced riders have a steeper incline than those designed for novices. Bicycle ramps made for competition (see [link])
More information1.1 The Pythagorean Theorem
1.1 The Pythagorean Theorem Strand Measurement and Geometry Overall Expectations MGV.02: solve problems involving the measurements of two-dimensional shapes and the volumes of three-dimensional figures;
More informationDeriving the General Equation of a Circle
Deriving the General Equation of a Circle Standard Addressed in this Task MGSE9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square
More informationTrigonometric identities
Trigonometric identities An identity is an equation that is satisfied by all the values of the variable(s) in the equation. For example, the equation (1 + x) = 1 + x + x is an identity. If you replace
More informationMathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh
Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions By Dr. Mohammed Ramidh Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because
More informationHow can we organize our data? What other combinations can we make? What do we expect will happen? CPM Materials modified by Mr.
Common Core Standard: 8.G.6, 8.G.7 How can we organize our data? What other combinations can we make? What do we expect will happen? CPM Materials modified by Mr. Deyo Title: IM8 Ch. 9.2.2 What Is Special
More informationLooking for Pythagoras An Investigation of the Pythagorean Theorem
Looking for Pythagoras An Investigation of the Pythagorean Theorem I2t2 2006 Stephen Walczyk Grade 8 7-Day Unit Plan Tools Used: Overhead Projector Overhead markers TI-83 Graphing Calculator (& class set)
More informationGeometry Vocabulary Book
Geometry Vocabulary Book Units 2-4 Page 1 Unit 2 General Geometry Point Characteristics: Line Characteristics: Plane Characteristics: RELATED POSTULATES: Through any two points there exists exactly one
More informationUsing Trigonometric Ratios Part 1: Solving For Unknown Sides
MPM2D: Principles of Mathematics Using Trigonometric Ratios Part 1: Solving For Unknown Sides J. Garvin Slide 1/15 Recap State the three primary trigonometric ratios for A in ABC. Slide 2/15 Recap State
More informationSPIRIT 2.0 Lesson: How Far Am I Traveling?
SPIRIT 2.0 Lesson: How Far Am I Traveling? ===============================Lesson Header ============================ Lesson Title: How Far Am I Traveling? Draft Date: June 12, 2008 1st Author (Writer):
More informationThe Pythagorean Theorem
! The Pythagorean Theorem Recall that a right triangle is a triangle with a right, or 90, angle. The longest side of a right triangle is the side opposite the right angle. We call this side the hypotenuse
More informationYear 10 Term 1 Homework
Yimin Math Centre Year 10 Term 1 Homework Student Name: Grade: Date: Score: Table of contents 6 Year 10 Term 1 Week 6 Homework 1 6.1 Triangle trigonometry................................... 1 6.1.1 The
More information2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle!
Study Guide for PART II of the Fall 18 MAT187 Final Exam NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will be
More informationPre-Calculus Unit 3 Standards-Based Worksheet
Pre-Calculus Unit 3 Standards-Based Worksheet District of Columbia Public Schools Mathematics STANDARD PCT.P.9. Derive and apply basic trigonometric identities (e.g., sin 2 θ+cos 2 θ= 1,tan 2 θ + 1 = sec
More informationTrigonometry Review Page 1 of 14
Trigonometry Review Page of 4 Appendix D has a trigonometric review. This material is meant to outline some of the proofs of identities, help you remember the values of the trig functions at special values,
More informationTERRA Environmental Research Institute
TERRA Environmental Research Institute MATHEMATICS FCAT PRACTICE STRAND 3 Geometry and Spatial Sense Angle Relationships Lines and Transversals Plane Figures The Pythagorean Theorem The Coordinate Plane
More informationMATH STUDENT BOOK. 12th Grade Unit 5
MATH STUDENT BOOK 12th Grade Unit 5 Unit 5 ANALYTIC TRIGONOMETRY MATH 1205 ANALYTIC TRIGONOMETRY INTRODUCTION 3 1. IDENTITIES AND ADDITION FORMULAS 5 FUNDAMENTAL TRIGONOMETRIC IDENTITIES 5 PROVING IDENTITIES
More informationAlgebra 2/Trigonometry Review Sessions 1 & 2: Trigonometry Mega-Session. The Unit Circle
Algebra /Trigonometry Review Sessions 1 & : Trigonometry Mega-Session Trigonometry (Definition) - The branch of mathematics that deals with the relationships between the sides and the angles of triangles
More informationMath 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas
Math 180 Chapter 6 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 6.1 Section 6.1 Verifying Trigonometric Identities Verify the identity. a. sin x + cos x cot x = csc
More informationModule Guidance Document. Geometry Module 2
Geometry Module 2 Topic A Scale Drawings 5 days Topic B Dilations 5 days Topic C Similarity and Dilations 15 days Topic D Applying Similarity to Right 7 days Triangles Topic D Trigonometry 13 days Just
More information8-1 Similarity in Right Triangles
8-1 Similarity in Right Triangles In this chapter about right triangles, you will be working with radicals, such as 19 and 2 5. radical is in simplest form when: 1. No perfect square factor other then
More informationSection 5.1 Angles and Radian Measure. Ever Feel Like You re Just Going in Circles?
Section 5.1 Angles and Radian Measure Ever Feel Like You re Just Going in Circles? You re riding on a Ferris wheel and wonder how fast you are traveling. Before you got on the ride, the operator told you
More informationMath 123 Discussion Session Week 4 Notes April 25, 2017
Math 23 Discussion Session Week 4 Notes April 25, 207 Some trigonometry Today we want to approach trigonometry in the same way we ve approached geometry so far this quarter: we re relatively familiar with
More information1 Trigonometric Identities
MTH 120 Spring 2008 Essex County College Division of Mathematics Handout Version 6 1 January 29, 2008 1 Trigonometric Identities 1.1 Review of The Circular Functions At this point in your mathematical
More informationMod E - Trigonometry. Wednesday, July 27, M132-Blank NotesMOM Page 1
M132-Blank NotesMOM Page 1 Mod E - Trigonometry Wednesday, July 27, 2016 12:13 PM E.0. Circles E.1. Angles E.2. Right Triangle Trigonometry E.3. Points on Circles Using Sine and Cosine E.4. The Other Trigonometric
More informationPythagorean Theorem Unit
Pythagorean Theorem Unit TEKS covered: ~ Square roots and modeling square roots, 8.1(C); 7.1(C) ~ Real number system, 8.1(A), 8.1(C); 7.1(A) ~ Pythagorean Theorem and Pythagorean Theorem Applications,
More informationGeorgia Standards of Excellence Frameworks. Mathematics. Accelerated GSE Pre-Calculus Unit 4: Trigonometric Identities
Georgia Standards of Excellence Frameworks Mathematics Accelerated GSE Pre-Calculus Unit 4: Trigonometric Identities These materials are for nonprofit educational purposes only. Any other use may constitute
More informationUnit Circle: Sine and Cosine
Unit Circle: Sine and Cosine Functions By: OpenStaxCollege The Singapore Flyer is the world s tallest Ferris wheel. (credit: Vibin JK /Flickr) Looking for a thrill? Then consider a ride on the Singapore
More informationDate: Worksheet 4-8: Problem Solving with Trigonometry
Worksheet 4-8: Problem Solving with Trigonometry Step 1: Read the question carefully. Pay attention to special terminology. Step 2: Draw a triangle to illustrate the situation. Decide on whether the triangle
More informationII. UNIT AUTHOR: Hannah Holmes, Falling Creek Middle School, Chesterfield County Sue Jenkins, St. Catherine s School, Private School
Google Earth Trip I. UNIT OVERVIEW & PURPOSE: will use pictorial representations of real life objects to investigate geometric formulas, relationships, symmetry and transformations. II. UNIT AUTHOR: Hannah
More informationMath 1205 Trigonometry Review
Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of
More informationChapter 8 Practice Test
Chapter 8 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In triangle ABC, is a right angle and 45. Find BC. If you answer is not an integer,
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More information13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. ANSWER: 2. If, find cos θ.
Find the exact value of each expression if 0 < θ < 90 1. If cot θ = 2, find tan θ. 8. CCSS PERSEVERANCE When unpolarized light passes through polarized sunglass lenses, the intensity of the light is cut
More informationExploring the Pythagorean Theorem
Exploring the Pythagorean Theorem Lesson 11 Mathematics Objectives Students will analyze relationships to develop the Pythagorean Theorem. Students will find missing sides in right triangles using the
More informationDuring What could you do to the angles to reliably compare their measures?
Measuring Angles LAUNCH (9 MIN) Before What does the measure of an angle tell you? Can you compare the angles just by looking at them? During What could you do to the angles to reliably compare their measures?
More informationChapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry
More informationHonors Algebra 2 w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals
Honors Algebra w/ Trigonometry Chapter 14: Trigonometric Identities & Equations Target Goals By the end of this chapter, you should be able to Identify trigonometric identities. (14.1) Factor trigonometric
More informationTHE PYTHAGOREAN SPIRAL PROJECT
THE PYTHAGOREAN SPIRAL PROJECT A Pythagorean Spiral is a series of right triangles arranged in a spiral configuration such that the hypotenuse of one right triangle is a leg of the next right triangle.
More informationc) What is the ratio of the length of the side of a square to the length of its diagonal? Is this ratio the same for all squares? Why or why not?
Tennessee Department of Education Task: Ratios, Proportions, and Similar Figures 1. a) Each of the following figures is a square. Calculate the length of each diagonal. Do not round your answer. Geometry/Core
More informationGeometry Problem Solving Drill 11: Right Triangle
Geometry Problem Solving Drill 11: Right Triangle Question No. 1 of 10 Which of the following points lies on the unit circle? Question #01 A. (1/2, 1/2) B. (1/2, 2/2) C. ( 2/2, 2/2) D. ( 2/2, 3/2) The
More informationMultiple-Angle and Product-to-Sum Formulas
Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite
More informationThe reciprocal identities are obvious from the definitions of the six trigonometric functions.
The Fundamental Identities: (1) The reciprocal identities: csc = 1 sec = 1 (2) The tangent and cotangent identities: tan = cot = cot = 1 tan (3) The Pythagorean identities: sin 2 + cos 2 =1 1+ tan 2 =
More information7.1 INTRODUCTION TO PERIODIC FUNCTIONS
7.1 INTRODUCTION TO PERIODIC FUNCTIONS Ferris Wheel Height As a Function of Time The London Eye Ferris Wheel measures 450 feet in diameter and turns continuously, completing a single rotation once every
More informationFigure 5.1. sin θ = AB. cos θ = OB. tan θ = AB OB = sin θ. sec θ = 1. cotan θ = 1
5 Trigonometric functions Trigonometry is the mathematics of triangles. A right-angle triangle is one in which one angle is 90, as shown in Figure 5.1. The thir angle in the triangle is φ = (90 θ). Figure
More information13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. SOLUTION: 2. If, find cos θ.
Find the exact value of each expression if 0 < θ < 90 1. If cot θ = 2, find tan θ. 2. If, find cos θ. Since is in the first quadrant, is positive. Thus,. 3. If, find sin θ. Since is in the first quadrant,
More informationT.3 Evaluation of Trigonometric Functions
415 T.3 Evaluation of Trigonometric Functions In the previous section, we defined sine, cosine, and tangent as functions of real angles. In this section, we will take interest in finding values of these
More informationConcept: Pythagorean Theorem Name:
Concept: Pythagorean Theorem Name: Interesting Fact: The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and
More information#9: Fundamentals of Trigonometry, Part II
#9: Fundamentals of Trigonometry, Part II November 1, 2008 do not panic. In the last assignment, you learned general definitions of the sine and cosine functions. This week, we will explore some of the
More informationθ = = 45 What is the measure of this reference angle?
OF GENERAL ANGLES Our method of using right triangles only works for acute angles. Now we will see how we can find the trig function values of any angle. To do this we'll place angles on a rectangular
More informationGeometry by Jurgensen, Brown and Jurgensen Postulates and Theorems from Chapter 1
Postulates and Theorems from Chapter 1 Postulate 1: The Ruler Postulate 1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1. 2. Once
More informationFigure 1. The unit circle.
TRIGONOMETRY PRIMER This document will introduce (or reintroduce) the concept of trigonometric functions. These functions (and their derivatives) are related to properties of the circle and have many interesting
More informationStudents apply the Pythagorean Theorem to real world and mathematical problems in two dimensions.
Student Outcomes Students apply the Pythagorean Theorem to real world and mathematical problems in two dimensions. Lesson Notes It is recommended that students have access to a calculator as they work
More informationMath 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b
Math 10 Key Ideas 1 Chapter 1: Triangle Trigonometry 1. Consider the following right triangle: A c b B θ C a sin θ = b length of side opposite angle θ = c length of hypotenuse cosθ = a length of side adjacent
More informationP1 Chapter 10 :: Trigonometric Identities & Equations
P1 Chapter 10 :: Trigonometric Identities & Equations jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 20 th August 2017 Use of DrFrostMaths for practice Register for free
More informationB. Examples: 1. At NVHS, there are 104 teachers and 2204 students. What is the approximate teacher to student ratio?
Name Date Period Notes Formal Geometry Chapter 7 Similar Polygons 7.1 Ratios and Proportions A. Definitions: 1. Ratio: 2. Proportion: 3. Cross Products Property: 4. Equivalent Proportions: B. Examples:
More informationUnit 8 Trigonometry. Math III Mrs. Valentine
Unit 8 Trigonometry Math III Mrs. Valentine 8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals.
More informationLesson 5: Area of Composite Shape Subject: Math Unit: Area Time needed: 60 minutes Grade: 6 th Date: 2 nd
Lesson 5: Area of Composite Shape Subject: Math Unit: Area Time needed: 60 minutes Grade: 6 th Date: 2 nd Materials, Texts Needed, or advanced preparation: Lap tops or computer with Geogebra if possible
More information7.1 INTRODUCTION TO PERIODIC FUNCTIONS
7.1 INTRODUCTION TO PERIODIC FUNCTIONS *SECTION: 6.1 DCP List: periodic functions period midline amplitude Pg 247- LECTURE EXAMPLES: Ferris wheel, 14,16,20, eplain 23, 28, 32 *SECTION: 6.2 DCP List: unit
More informationPythagorean Identity. Sum and Difference Identities. Double Angle Identities. Law of Sines. Law of Cosines
Review for Math 111 Final Exam The final exam is worth 30% (150/500 points). It consists of 26 multiple choice questions, 4 graph matching questions, and 4 short answer questions. Partial credit will be
More informationcos sin sin 2 60 = 1.
Name: Class: Date: Use the definitions to evaluate the six trigonometric functions of. In cases in which a radical occurs in a denominator, rationalize the denominator. Suppose that ABC is a right triangle
More information13-3The The Unit Unit Circle
13-3The The Unit Unit Circle Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Find the measure of the reference angle for each given angle. 1. 120 60 2. 225 45 3. 150 30 4. 315 45 Find the exact value
More informationRosa Parks Middle School. Summer Math Packet C2.0 Algebra Student Name: Teacher Name: Date:
Rosa Parks Middle School Summer Math Packet C2.0 Algebra Student Name: Teacher Name: Date: Dear Student and Parent, The purpose of this packet is to provide a review of objectives that were taught the
More informationGeometry. Teacher s Guide
Geometry Teacher s Guide WALCH PUBLISHING Table of Contents To the Teacher.......................................................... vi Classroom Management..................................................
More informationTrigonometry Review Tutorial Shorter Version
Author: Michael Migdail-Smith Originally developed: 007 Last updated: June 4, 0 Tutorial Shorter Version Avery Point Academic Center Trigonometric Functions The unit circle. Radians vs. Degrees Computing
More informationAlgebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position.
Algebra and Trig. I 4.4 Trigonometric Functions of Any Angle In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. IN this section we
More information( for 2 lessons) Key vocabulary: triangle, square, root, hypotenuse, leg, angle, side, length, equation
LESSON: Pythagoras Theorem ( for 2 lessons) Level: Pre-intermediate, intermediate Learning objectives: to understand the relationship between the sides of right angled-triangle to solve problems using
More informationEstimating Tolerance Accuracy (Rounding, including sig. fig.) Scientific notation
S3 Pathways for learning in Maths Pathway 1 (Lower) Pathway 2 (Middle) Pathway 3 (Upper) Targets Complete coverage of level 3 experiences and outcomes in Mathematics Cover level 4 experiences and outcomes
More informationSimilarity and Ratios
" Similarity and Ratios You can enhance a report or story by adding photographs, drawings, or diagrams. Once you place a graphic in an electronic document, you can enlarge, reduce, or move it. In most
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationChallenging Students to Discover the Pythagorean Relationship
Brought to you by YouthBuild USA Teacher Fellows! Challenging Students to Discover the Pythagorean Relationship A Common Core-Aligned Lesson Plan to use in your Classroom Author Richard Singer, St. Louis
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1316 Ch.1-2 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Find the supplement of an angle whose
More informationLesson 6.1 Skills Practice
Lesson 6.1 Skills Practice Name Date Soon You Will Determine the Right Triangle Connection The Pythagorean Theorem Vocabulary Match each definition to its corresponding term. 1. A mathematical statement
More information